A few years ago, the chiral algebra (CA) bootstrap was proposed for computing loop-level amplitudes in non-supersymmetric gauge theories. The main limitation of this method is that it applies only to twistorial theories rather than to QCD, the theory relevant at hadron colliders. In this talk, I will nonetheless show how the CA bootstrap can still help in the computation of certain two-loop QCD amplitudes. I will begin by reviewing the basics of twistorial theories and the CA bootstrap. I will then explain some consequences that these twistorial theories have on one- and two-loop gauge-theory amplitudes. Finally, I will show that the two-loop all-plus-helicity n-gluon amplitude of QCD is given by CA bootstrap results and lower-loop amplitudes, circumventing the need to compute any two-loop integrals.